Axon Guidance :

I created this animation to  depict the axonal growth .


Unlike other sensory systems, olfaction is quite complex because of  its
large number of receptors. These receptors are placed in a random fashion
in the  nasal epithelium,  where  each kind (defined by their receptors)
of neurons has to project  their axons to a specific part of the olfactory-bulb.Once projected
( which takes few days), the axons usually live for about a month and
is replaced by a new axon dynamically.  This complex dynamical process process is
not just guided by the  chemotaxis, or the guiding
axons, there are biological evidence suggesting axonal interactions. We
(with M. Zapotocky, D. Chaudhuri and P. Borowski)  tried to capture the
phenomena in a simple model of interacting directed random walk in $1+1$ dimension.
The path of the random walkers were modelled  as axon-shafts and the growth cone
were modeled by the walkers. In our model, the  growth cone interacts with the
shaft and tried to be closer to the shaft of the same kind. Turn-over  and birth were
introduced to model  the dynamical process.

 Interestingly we find that the system takes unreasonably longtime (in units of
average life time of a axon)  to reach the steady state, independent of details of
the rule. We have naive understanding of this long relaxation time. During growth,
certain kind of scaling is observed  at  a distance sufficiently far away  from
the epithelium which allow us to  calculate the change in number of fascicules
in time and in distance (measured from the epithelium). Once the microscopic
 parameters are understood, we hope to
make specific quantitative predictions  which can be verified  in experiments.